THE CURRENT CARRYING CAPACITY OF POWER CABLES IN TUNNELS

Birron M. Weedy

University of Southampton,England.

Abstract-The capital costs incurred in the provision of power cir-cuits through underground tunnels is such as to make further efforts inthe refinement of thermal rating computations worth while. In the pres-

ent work steady state and transient temperature calculations are per-

formed via thermal network models of the system. Heat transfer byradiation is separately accounted for and heat transfer by free or forcedconvection determined by means of new empirical formulae establishedexperimentally on groups of cables.

From the results various aspects of thermal performance are dis-cussed. These include the effects of the following; free or forced con-

vection, tunnel depth and cross section, diameter of cables and type ofgrouping, outlet and inlet air temperatures, ambient changes, soil dryingout, transient temperatures resulting from application of load and lossof air flow.

INTRODUCTION

The installation of power cables in tunnels provided for other pur-

poses is well established. In generating stations short tunnels are oftenused to convey a large number of cable circuits. Long tunnels are builtor existing tunnels adapted solely for the purpose of carrying majorE.H.V. transmission circuits which for various reasons cannot be carriedoverhead. River crossings are obvious cases where tunnels would be usedeither for technical or amenity reasons. The cost of such installations isvery considerable and it is desirable to optimize as far as possible thecurrent carrying capacity, groupings, and number of circuits to be in-stalled to meet a given transmission capacity.

In the pre-determination of thermal conditions in tunnels for bothforced and free (natural) convection the following factors are relevent:-

(a) the maximum conductor temperature should not exceed thestipulated value for the type of cable, e.g. for oil-filled cables, 85°C;

(b) the maximum air temperature should not exceed a stipulatedvalue, e.g. 35°C for reasons of working personnel in the tunnel;

(c) the range of inlet temperatures;(d) when forced cooling is provided the effects of loss of coolant

flow over a period of time;(e) the effects on the cable ratings of the tunnel depth, cross sec-

tion and environment;(f) whether to base ratings on the safe premise of free convection

only or to assume some degree of forced convection due to naturaldraughts even if fans are not provided.

In a companion paperl the present authors give empirical formu-lae for the prediction of the heat transfer coefficients at the surfaces ofcables in various groups in tunnels. It is intended here to discuss compu-

tational methods by means of which cable and air temperatures andother relevant quantities may be predicted. Detailed investigations intothe rating of cables in tunnels have been given by Giaro2, Kitagawa3,Burrell4, Germany5 and Whitehead and Hutchings6. These are based on

heat transfer evaluations from equations established for mechanical andchemical installations. It is hoped that the present methods which stemfrom measurements on cable assemblies will provide a more accuratesystem of assessment. In the present study a circuit is always assumedto comprise three, single conductor, cables.

Paper T72 505-6, recommended and approved by the IEEE Insulated Con-ductors Committee of the IEEE Power Engineering Society for presentation at theIEEE PES Summer Meeting, San Francisco, Calif., July 9-14, 1972. Manuscriptsubmitted January 6, 1972; made available for printing May 24, 1972.

Thermal Characteristics of Tunnels

Before considering the thermal processes inside a tunnel the tem-perature-rise characteristics of the tunnel walls with an internal heatflux uniformly applied both axially and radially is of interest. Thisstudy will indicate the effects of depth and changes in ambient tempera-ture on the tunnel temperatures. For steady state and transient tempera-ture calculations relatively simple analytic expressions may be appliedprovided the depth is sufficient. However it is desirable to include suchfeatures as small depth, difference in thermal resistivity of the wallscompared to that of the environment and effects of drying out of theenvironment. To facilitate this a finite difference method and digitalcomputer are used. Unit axial thickness is considered and the tunnel andarea around it, up to the soil surface, are subdivided into a two dimen-sional finite-difference mesh. For the steady state calculations successiveover-relaxation is used to compute the nodal temperatures and for thetransient calculation the simple explicit step-by-step method. The largerthe area encompassed in the study the more accurate the results. How-ever, the number of nodes increases very rapidly with area and a com-

promise is made to keep computer storage and operating time to withinreasonable limits.

0

)- / CFC|~~ ~~ ~~~~~~~~L- *xrtIZ

Time Moiortf5

Fig. 1(a). Temperature rise of tunnel walls.Depth > 6m, gsoil, 1 20°C cm/W.

In Fig. I a square tunnel of side 3m and situated at various depthsis considered. The transient temperature rise of the tunnel walls when a

constant internal heating is suddenly applied is shown in Fig. l(a).Prior to this heating the tunnel and surroundings are at ambient. Thiscurve applies for all parts of the tunnel walls for depths greater than6m. Fig. 1(b) applies to the same tunnel with constant internal heatingbut with a 20°C step function of temperature change on the ground sur-

face. The depth is now less than 3m. Even at this low depth the suddenchange in ambient takes a considerable period to be effective by whichtime the general seasonal ambient will have changed. In Fig. 1(c) a veryshallow tunnel is considered with the soil above the side walls replacedby a concrete slab of 0.6m thickness. Again constant heating is appliedwith a step function of change in ground temperature. Only with thisinstallation are surface changes immediately effective at the tunnel walls.

Although the situations and events shown in Fig. 1 are idealisedthey indicate the very considerable time lags of tunnels at reasonabledepths for changes in ground (ambient) conditions. The situation is such

298

H.M. El Zayyat

face to air (convective), cable surface to walls (radiation), air to tunnelwalls (convective). The formulae used for computing heat transfer coef-ficients are given in reference 1. The effective thermal resistance of thetunnel walls is calculated assuming a circular tunnel and the effectivesoil resistance by the traditional (Kennelly) formula. The soil transientcharacteristics may either be obtained by the use of the exponentialintegral formula;

0(t) - {-Ei (_ D2 "' + Ei (- t)

Fig. 1(b). As (a) plus ambient change.Depth = 3m.

or by dividing the soil into annular cylinders with the assumption ofuniform radial flow and representing each cylinder by its radial resist-ance and capacitance per unit length of tunnel. Thermal resistance ofthe soil in the axial direction is neglected. Generally, radial thermal re-.sistances per unit length are calculated by the expression,

Dn+n and capacitances per unit length by,n

4 (D 2 - D 2) pcT n+1 n p

The heat transfer resistance for n cables, R n=rDh per unit length.The thermal resistance between the moving air and tunnel wall is givenby3; 7.22 0C/W

(Dt Vm) per cm length (2)m~~~~~~~~~~~~~~~~()

IO-

Q,p + Qgrwa J,

QC

o C a

Fig. 1(c). Depth 0.6m; top of tunnel is concrete (g = 60°C cm/W),thickness 0.6m.

that in winter the inlet cooling air, which would normally be at ambient,will over part of the tunnel length receive heat from the tunnel wallswhich are still appreciably above ambient.

Thermal Model of the System

The process of temperature prediction is facilitated by the use ofequivalent thermal lumped-constant networks which represent the ther-mal fields. Heat is evolved through the losses in the cable conductor(I2R), dielectric and sheath and transferred to the air by convectionprocesses and to the walls by radiation. Heat may also be removed fromthe air by forced convection heat transfer from the air to the tunnelwalls.

A cable may be represented by a thermal ladder network formedof thermal resistors and capacitors with the various losses injected at theappropriate nodes. This assumes that the heat flow is uniformly radial,i.e. that the sheath is isothermal. The degree of subidvision of the die-lectric determines the accuracy. For reasonably long thermal transientsthe cable R-C ladder network may be simplified to a simple 7r network.

Heat transfer resistances are directly represented, e.g. cable sur-

(a.- QQ)Fig. 2. Steady state thermal network forced cooling.

W7a Vs

\dcutJ'

Fig. 3. Steady state thermal network - free cooling.

The equivalent steady-state thermal network per unit longitudinal lengthis shown in Fig. 2 for forced convection and in Fig. 3 for free or natural

convection.

Steady-State Thermal'Analysis Forced Cooling

The temperature rise of the air Tx at distance x from the inlet isobtained from the thermal balance equation derived from Fig. 4. It maybe shown that,

299

20

cQ

awL

(%W.

W1

0

(1)

2o

T = T + Q R - {Q R - T. + T I eX W CC CC I. W

where XRpc- AVRc PpA

Tw

roow riTIxd_ _Ti _X TH+e1t

Fig. 4. Analysis of air temperature.

In reference 1 the values of the ratio of convective to radiant heat tr:fer for a variety of cable sizes and groupings are given. F is the r;

Convective heat ) andTotal dissipated heat

Qc U nF (12R + Wd + WS)Q U 1 - F Qr c

From Fig. 2,

Tw - Te + (Rwall + Rsoil Mr + Qp)T - T

and Q Ux wp Rc

From the above equations,

Steady-state Thermal Analysis - Free Cooling

(3) From the thermal network represented in Fig. 3 the followingequations are obtained:

lwT - T -(I2R + Ad) R -(I2R + Wd + Ws)(Rb serS C 2 d d sliRser)

T UT + Q Rs w r r

w e r wall soil

T = T - Q R 1a s c c

(13)

(14)

(15)

(16)

and from reference 1,

RUr1

3 2 T + T2 1T)Kr.An a(aTs aT aTw +aTs a s

ans- Then equation (14) becomes

atio Ts = Tw + n (1 - F) (I2R + Wd + WS)(1

(4) Kr An a(as3 aTs aTw +aTsaTw2 aTw)(17)

Inserting equations ( 13) and ( 15) in equations ( 17) and after successivereductions, the following polynomial equation is obtained:

(5) ItR4 + (Al) IbR3 + (A2) P4R2 + (A3) I2R + (A4) U 0(18)

The coefficients A1, A2, A3 and A4 are defined in Appendix I andequation (18) is solved by the Newton-Ralphson method.

Tw Te+(Rwall+Rsoil ){n(l-F)(I2R+Wd+Ws)+nF(I2R+Wd+WS)-Xx Ti _Ax Tw -Xx(l-e ) + Re - -e }

c c

Transient Thermal Analysis(6)

Tw = Bl + B2 I2 (7)

where B 1 and B2 are coefficients depending on the system parameters.From equations 6 and 3

TU= C1 + I2C2 (8)

where -Xx + l-XxC = T. e- + (l-e ) (B1+ nF (Wd + Ws) Rc) (9)

and C2 U (B2 + nF R Rc) {I - eXx (10)

Also, Ts - Tx + Itc Qc

U1 +IC + nF (I2R + Wd + Ws) Rc1.Ts= C +

2C12

and WT = T (I2R + -) R -(I2R + W + W )(Rb + R ) (12)5 C 2 d d s ser

If Tc is stipualted to be a certain temperature, e.g. 85'C, then fromequations 11 and 12 the current per cable (I) can be calculated. Also bysubstituting the value of I in equations 7, 8 and 11 Tx, Tw and Ts are

determined.

Although perhaps of secondary importance to steady state consid-erations the transient performance is important to determine the effectsof limited-period overloads and the changes which occur when forcedcooling ceases and the cooling regime reverts to free cooling for a periodof time. For the times under investigation sufficient accuracy is ob-tained in the thermal model by subdividing the soil around the tunnelbe concentric rings each annular volume of soil being represented by a

lumped thermal resistance and capacitance.In order to reduce the complexity and computation times each

cable is represented by a "pi" section as suggested by Van Wormer7 inwhich the thermal capacitance of the dielectric is split between the con-ductor and sheath nodes as indicated in Appendix 2. As for the steadystate analysis heat balance equations are established at each node but inthis case at each time step. The size of the time step is governed bymathematical stability considerations. A detailed treatment is given inAppendix 2.

Results

Because of the large number of variables involved it is difficult togive a comprehensive account of the results obtainable by means of theabove methods. Such variables are; air velocity, tunnel cross section andlength, cable size and grouping, tunnel depth and air inlet temperature.An attempt will be made however to quote sufficient relationshipsbetween these quantities to enable general trends and effects to be seen.Discussion will mainly centre around groups of 132 kV, 290 mm2 con-ductor area, oil filled cables. As the heats evolved per unit length fordifferent sizes of cable conductors at their appropriate ratings are roughlycomparable, many general conclusions may be drawn from this type ofcable.

300

All the results presented are computed using the above methods.In order to provide a reference point for all the results and to avoid con-fusion the following assumptions are made:

(a) all ratings and corresponding air temperatures relate to asteady state temperature of 85°C for the conductor of the hottest cableof the group.

(b) all the air velocities quoted are those at the tunnel outlet.Figs. 5, 6, 7, 8 and 10 refer to a very shallow tunnel with a cover

comprised of 0.5 m of concrete and approximately 1 m of soil.

Z2-ca

k)

2~I-.Id

IT

5O 1

J 3-0514/S

3 11000

tV

aQI..Q

Ii)

TL..- 3 50c

- 320-C

}3°

3 6

AiP, VrLocrry /5

Fig. 7. Ratings and air velocity. 132kV, 290mm2 cables in trefoil,--- 2 circuits, 6 circuits, tunnel length 0.75km, diam-eter 2.7m.

[5LrtqT Oic Yurqt4Fw- k,vj

Fig. 5. Ratings at vario2us inlet temperatures and air velocities. Six,400kV, 1935mm circuits in flat formation, spacing 25 cm.

U4%0

Id

(of

10.4

"I1

1..

0 15k

Fig. 6. Ratings at various inlet temperatures. Six, 132kV, 290mm2 cir-cuits; flat formation touching; Dt2.7m.

AiR VELOC1r.Fig. 8. Air exit temperature. --6 circuits, 2 circuits, 132kV,

290mm2, trefoil. Tunnel length 0.75 km, depth > Om, diameter2.7m.

In figures 5 and 6 the current ratings of groups of cables with dif-ferent lengths of tunnel and inlet temperatures are shown. The conduc-tor temperature is directly dependant on the air temperatures and so

the decrease in allowable current with tunnel length is less marked atthe higher air velocities. In Figs. 7 and 8 the variation of current and airtemperatures respectively with air velocity are shown. It is seen that themajor gain in rating is obtained up to a velocity of 1.5m/s. Further in-crease in velocity up to 6m/s produces only a 20% increase in currents.This, however, assumes no direct limitation to the value of air tempera-ture. If the exit air temperature is limited to 35°C with the above rat-ings the air velocities greater than 1.5m/s would be required especiallyat the higher inlet temperatures. This latter aspect is again illustrated in

Fig. 9 where the dependence of air exit temperature on number of cir-cuits is shown. It is seen that only two circuits (six single-conductorcables in all) are permissible at 1.5 m/s whereas six circuits with an air

301

18

TI.

_-.-

-- - - zOC

5O'C

0

70r

750 .

350

5oC

51X CIRCUITS

2 3 4

r%10 OF Cl c ITS

Fig. 9. Air exit temperature and number of circuits.trefoil; tunnel length 0.75km, daimeter 2.7m.

, o~~~~14-6.

orPT" Or TuV r-*. (i)

2Fig. 11. Ratings and tunnel depth. 6, 132kV, 290mm2 circuits. Cables132kV, 290mm2, in flat formation and touching Free convection.

FLATr - Touct l vc

0 FA.Amr- ?ZSC.-% SPACAW&

~ ~ A,vLC Y /

TgLzF oiA..

700

AiRt VCLLCGIT-Y

Fig. 10. Effect of groupings. TIN 50C, tunnel length 0.75km, diameter2.7m, 132kV, 290m2, 6 circuits.

800

zw4

I C.C.

- - - ~~~~~~~~-- -- - I,CL

lCoCC.t

400200 300 4oo ,oo

TuNNCLN D9A rct ( n)Fig. 12. Rating and tunnel diameter - free convection. Depth

2.4m,--- Depth 1 3.65m, 1 32kV, 290mm2, flat formation,spacing 25cm.

velocity of 6 m/s bring the exit air temperature to just above 35°C. Itshould be remembered however that there is a decrease in the currentrating according to number of circuits in use based solely on the cri-terion of conductor temperature. With free convection (deep tunnel)the cables must be derated to keep the air temperature at 350C withonly one circuit.

The effect of different cable formations are shown in Fig. 10.From a current-rating viewpoint the horizontal or vertical spaced for-mation is preferred. Although trefoil touching may have installation

advantages its heat transfer properties are the worst obtainable, addedto which the sheath losses in large conductor cables may be substantial.

With free convection the effects of tunnel depth on current ratingsare pronounced. This is illustrated in Fig. 11 where it is seen that a con-

siderable reduction in current results with increasing depth of tunnel.The rate of decrease becomes small for depths greater than 15 m. In-crease in tunnel cross sectional area with free convection cooling showsonly a significant increase in circuit rating with several cable circuits as

shown in Fig. 12.

Transient temperature changes are shown in Fig. 13 which includesthe conductor temperature rise when the air flow ceases for a few hours-and cooling is via free convection. This graph relates to two, 400kV,1935mm2 circuits carrying 2500A in a lkm tunnel of diameter 3.6m.The air velocity and inlet temperature are 6m/s and 25°C. It is seen thatthe rise in conductor temperature is rapid and that it remains above100°C for approximately 15 hours, the air flow recommencing after 9hours.

With up to six circuits the temperatures in the soil adjacent to thetunnel are only 50°C or above with free cooling. At this level drying outof the soil may occur with subsequent increase in thermal resistivity.For example with two, 290 mm cable circuits in a deep tunnel the walltemperature is 50°C for an ambient of 1 0°C. With six such circuits thewall temperatures exceed 500C for all depths below 1.4m for ambienttemperatures of 5°C and above (2.7m diameter tunnel). The effect ofthe drying out of rings of soil concentric to the tunnel containing a

single circuit of 965 mm2 cables on the current rating is given in Table

302

5o

0-

0.

w] !

ki

F-

30

701

0

(Xw

0

85Iz LK t | F toss OF Aig FRowz; |I*^ r |f Fs0

(0

0 30 90

7G s, Hovr.s

Fig. 13. Loss of air flow.

1. With forced cooling the wall temperatures are much lower and theonset of the drying-out process less likely.

Table 1

Moisture Migration and Ratings

Rating (A)

Confi uration965mm cables,diameter 91mm

Ring around tunnelHonogeneous (g - 300) of radial

thickness:-0.2m 0.5m lm

without fans some natural ventilation will exist although without meas-urements in existing installations this is difficult to quantity. Air flowmeasurements in existing tunnels would be of great value in building upknowledge on which to calculate ratings on a less pessimistic basis.

Whereas progressively smaller gains in current capacity result fromhigher air velocities (based on 85°C conductor temperature), highervelocities are required to keep the air exit temperature below a pre-scribed maximum, e.g. 35°C. With free convection such a restriction onthe bulk-air temperature imposes a severe limit to transmission capacitythrough a tunnel.

With forced-convection cooling the tunnel depth below the surfacehas negligible effect on current ratings. However with free convectiondepth has a marked influence. For depths greater than 15 m furtherchanges in depth cause little change. Transient temperature changes ofthe tunnel walls resulting from changes in heat input are increasinglyslow moving as the depth increases. In particular for deep tunnels,changes in ambient or ground surface temperature are manifested sig-nificantly at the tunnel walls only after the passage of months. As thetemperature of the inlet air will normally approximate to the ambientit is possible for the tunnel walls to be at a temperature typical of oneseason and for the inlet air temperature to be typical of the succeedingseason. This will materially effect calculations of cable rating.

Transient changes in cable conductor temperature when the heattransfer mechanism changes from forced to free convection, e.g. due tofan failure, are relatively fast. In a typical installation a stoppage of airflow for approximately 9 hours followed by a resumption of forcedcooling results in the temperatures shown in Fig. 13. In recent years ithas been assumed that drying out of soil may occur at soil temperaturesin excess of 50°C. With an installation of 6 circuits of 965 mm2 cablesand an inlet temperature of 15oC the tunnel walls reach 50°C with anair velocity of 1.5 m/s for a tunnel length of 1.5 km. However for thesame tunnel with only free cooling the walls attain almost 50°C withonly one circuit. If the environment of the tunnel is such that dryingout is possible (e.g. with clay) then with free cooling this may cause arestriction to transmission capacity, with forced cooling it is unlikely.

1530 1490 1350

1415 1370 1240

Tunnel depth 2m, diameter 3.6m;g in 0C cm/W

CONCLUSIONS

Thermal networks in conjunction with new experimental data on

cable heat transfer have been used to predict the ratings of cables intunnels. An essential part of the computation process is the separatetreatment of radiation heat transfer. For example with a single 62.5 mmdiameter cable, 10% of the total heat is transferred to the walls by radi-ation with an air velocity of 6 m/s; at 1.5 m/s this increases to 28% andwith free cooling to 42%.

A major problem is the predetermination of air velocities in thevicinity of the cables, i.e. near the walls and floors as opposed to thebulk air velocity. This is especially so where natural draughts are presentas distinct from fan induced flow. Also the increased possibility of a

severe fire and the corresponding cost and complexity of adequate pre-

cautions is a pertinent factor when forced cooling is provided. In viewof these factors cable installations are often designed on the basis offree convection only. The results obtained in the present work show thisto give much reduced transmission capacity compared with that ob-tained for the lowest forced-convection velocities. It is likely that even

APPENDIX I

Free Cooling - Calculation of Temperatures

Coefficients of the polynomial equation (18)

(Al) = (Bl + B2 + B3) /B4, (A2) (B5 + B6 + B7 + B8) /B4,

(A3) - (B9 + B1O + Bll)/B4, (A4) = B12/B4

where

Bl Co.CC2.C24.CC5, CO = A n.K .a,

CC2 = 2 (C3d0C4 - Cld'C2) C3d = n (1 - F) Wd

(wallR soil) + Te + 273, C3 =Wd.4 + Te

04 =n (I F)(Rwall R1soil)cId 273 +

Tc d

(Rd/2 Rser + ), Cl c Wd (Rd,2 ser Rb),C2 - Rd + Rser + Rb C24 - C4 - C2, CC5 = C2 + C4,B2 = CO.03.13.0C5, 03 = C22+ C42, C13.C1 + C3 + 546,

B3 = CO.CC3.C24.CC4, CC4 = C3 C1, B4 =CO.CC3.C24.CC5,B5 = CoX03*13'0C4, B9 = CO.CCl.C13.CC5 + (1 - F)n,

Bll = CO.CC2.Cl3.CC4, B10 = CO.CCl.C24.CC4,B12 = ColC13.CCl.CC4 + (1 - F) Wd.n, B6 = CO.2.13.CC5,B7 = COX.C2.CC4.C24, B8 = CO.MCC13.CC4.

303

Spaced-horizontaland vertical 1560

Touching-horizontal 1450and vertical

After calculating I; TS, TW and Ta can be calculated from the fol-lowing reduced forms:

T = C1 - C2 I2R

w 3 4

(1-F) (I2R + Wa)T -T d-{2a s 7wKG (aD3gp2c )P

pUK

C1, C2, C3 and as defined above, G and p are the constant and expo-nent of the free convection formula: NNU,D = G(NGrNPr)P.

For stability in the computational process the coefficients of the tem-perature terms must be positive; i.e.,

H + Ho.I1- Ccd 1) At must be positive and hence,

cdCd

At < H + Hd oil

CdsFor the sheath node: (T -T h) Hd+ Wd/2 (Tsh-Tsh)+ (Tsh T ser) Ht

and, (H +H )At H At T .HAt wd At

T T { d t +T - + ser t +sh sh Cds c Cs Cds ds

APPENDIX II

Transient Thermal Equations

The overload capacity of cables is goverened by the conductortemperature rise/time characteristics. In formulating the transient ther-mal equations the following assumptions are made.

(a) uniform radial flow of heat from the cable surface.(b) the soil round the tunnel is subdivided into concentric annu-

lar volumes each represented by a thermal resistance and capacitance,i.e. RS, CS,; --- RSn CSn

(c) the cable is represented by the Van Wormer 7r section. In this,part p' of the thermal capacitance of the insulation is placed at theconductor node and (1 - p') allocated to the sheath node, where,P'= I _ ,2__1

D D22kn(-0) o

Di Di2The heat-flow equations obtained for forced cooling from Fig. 14 are asfollows:

RIw

Co.I

Rc

wv1

6-

TJs

Fig. 14. Thermal network for transients - forced cooling.

Conductor node;

Wd CW + =(Tc'-Tc) ct+ (Tc oil) oil C

c sh d

Substituting for Wcu =I 1 + 0.00393 (Tc - 20) and re-arranging, thenew temperature after time step At is,

where, At < Ha +H

For the serving; (T h T ) Ht - (T ser

+ ser srseri.e., (H +H )At H At

T I = T (H-t ser I +

tT

ser ser Cser Cser sh

and,

C- Tsser

ser At

H .Atser

+ Cser

CAt<

serAt H + H

t ser

For the cable surface;

Ta + Ta(T -T1) H =(T- al a2ser s ser = 2 ca rQ

and, H H

Ts Tser H +H al 2 (H +H )ca ser ca ser

Hcaa2' 2(H +H )ca ser

- {(irmD- 2A).a.Kr.L.(aTs4 - aTw4)1}/(Hca + Hse)

where Am is half the mutual radiant area between the hottest cable inthe power circuit and the other two cables in the circuit.

Since the tunnel in forced cooling is divided into several axialcylinders the average of the inlet and outlet temperatures of the con-sidered region is taken and,

Ta +T 2Ca(T - al a2 C a i + V' (Tt-T)al

(Tal +Ta2 Tw) Haw+

-

2

(H +H . )At H AtT ' = {1- d oil } T + T

cd cd

H . .Atoil-+ TC cd oil

+ I2R {1 + 0.00393 (T - 20)1 CAtcd

The energy flow rate involved in the transient temperature rise of theair within the elemental volume over the time step At is,

C(T ' -T )a

a2 a2 At

304

Vs Q, P%r

3e Rwau Aso Rsn Rs To-NA-^ N^VV%p-

rs., fo. KL TS I T-anIV

tlyAI -L -L I ICier ( C4, Cw CSI- IC sit

I I I 91-

This last term should contain the temperatures Ta2 + Tand Tl + a2

2 ad 2instead of Ta2, and Ta2 respectively, but since the average air tempera-ture contains both inlet and outlet temperatures, the use of the averagetemperatures would require a solution for two unknown temperaturesafter the time step. Hence the downstream values are used to representthe average. Therefore,

Ta2' Ta {1a2 a2

+T

(2V' + nH + H ) At

* ~~2Ca

(2V' - n.H - H ) Atca a2Ca

Air velocity 3 m/s, air inlet temperature (Tl) 1 50C. Thermal param-eters (see equation 3 to 12) - all thermal resistances and losses are percm length of system.

Rd = 5 ln 2:54 ) = 48 deg. C/W

Rsh = 29 ln ( 4.674 ) = 0.044 deg. C/W

Similarly Rb + Rser = 16 deg. C/W. The sheath loss is assumed to bezero, i.e. Ws = 0. Wd = 0.0416 W/cm and 12R (at 85°C) = 0.96 X10-6 I2W/cm.

n.H. .At+ T ca + Tw H Cs C w aw'

a a

2Cand, At < 2V' + n.H + H

ca aw

From the coefficient of Tal term is obtained the space criterion forstability, i.e.,

i.e. n.H + H < 2V'-'ca aw

But: Hca 7rDhc. L, where L = length of subdivision and,

7.22.LHaw 075

(D V )tm i

and V' = A.V.P.cp

I 2.A.V.p.cp

{wrDn h + 7.22 IC (D mtm m

Thus

T?innel wall node:(T+Ta2) ~~~~~(T'-T )x2( al T'HT+a w wI

2 Tw1 aw R1Qr1

But Qr = An'Kr a (aTs4 - aTw4), hence Tw'. Other nodal equationsare derived in a similar manner. The lowest value of At is chosen to ob-tain the overall stability criterion for all nodes.

For free cooling the thermal network is modified such that theresistive part corresponds to Fig. 3. The heat flow equations for Tc'.Toi, Tser', Twl', Ts5', and T 2' are the same as those developed in theprevious case.

APPENDIX III

Example to illustrate the calculation of the forced cooled rating ofsix cable circuits each in trefoil formation.

Specification - Cables, 132 kV, 290 mm; parameters given inAppendix V.

Tunnel - 0.75 km in length; rectangular cross-section of mean

hydraulic diameter 2.7 m; shallow installation with 0.5 m concrete roofplus approximately 1 m of soil (g = 90 deg. C cm/W.)

The effective thermal resistance from inside tunnel wall to groundsurface (ambient to 1 5°C) is 8 deg. C/W per cm length (Rwall + Rsoi1).The tunnel approximates to the situation relevant to Figs. 6, 7 and 8.

R = 7.22C (D V ) 0.75t m

where Dt and Vm are in m and m/s respectively.

i.e. Rc= 1.51 deg. C/W.

R p A V= 3.3 x 10-5 per cm;c p

Xx = 2.47

and e4X = 0.0846.From reference (1), for 6.25 cm diameter cables in trefoil, F=

0.68.Number of cables n = 6 X 3 = 18.From reference (1) for cables in trefoil, for hottest cable,

N = hD = 0.07 (Ne)0*65

For v = 3 m/s, NRe = 12400 and NNu = 32 h = 12.8 X 10-4 W/cm2 perdeg. C and henice RC = hw 6 25 = 3.97 deg. C/W and R 1 - 3987 = 2.2deg.Candhe cRa -f 1.3 c 18deg. C/W.

From equation 6

Tw 1 0.0846x 8) = 15+8 F 18 x 0. 32 (12 0.96

+ 0.042)1.51 L106

+ 18 x 0.68 (I2 .9 + 0.042) 0.9154 + x 0.0846'110FTX

Hence Tw = B1 + B2 12 = 13 + I2 36.2 x 10-6 (0C)

Also from equation 8, Tx = C1 + 12C2, where

C= 15 x 0.0846 + 0.915 x 1.51 (13 + 18 x 0.68

x 0.048) = 14.0

C = 36.2 + 18 x 0.68 x 0.96 x 1.51) 0.9154 =2 106 106

= 49.5 x 10-6

T = 14 + I2 + 18 x 0.68 x 2.2 (I2 x 0.96 x 0.042)S 106 106- 15.13 + 5 12

106Also,

T = 85 - (I2 + 0.021) 48 - (I2 9 + 0.042) 16106 106

12= 83.4 _ - x 60

106

305

From which, I = 720A, and Tx (air outlet temperature) = 390C Theseresults can be obtained from Figs. 7 and 9.

APPENDIX VI

Table 3

Constants of Cable Materials

APPENDIX IVMaterial

Thermalresistivity°C cm/W

Heat Transfer Formulae

The practical determination of new empirical formulae, for heattransfer from cable surfaces in tunnels is discussed in reference 1. Theresults relevant to this paper are summarized below.

Forced Convection

NNu = C(NRe)0 65 for all formations. Values of C (hottest cableonly); flat touching 0.086; vertical touching 0.086; vertical spaced (spac-ing/diameter > 2) 0.115; horizontal spaced (spacing/diameter > 2)0.1 1 5; trefoil 0.07.

Free Convection

NNu,D F (NGr,D Npr)P Valuetions (hottest cable only) are as follows

horizontal spacedvertical spacedhorizontal touchingvertical touchingtrefoil

The mutual radiation area (Am) IX Diameter for trefoil formation.

APPENDI

Table

Particulars of

Particulars

A.C. resistance of theconductor per unit lengthat 200C (micro ohm/cm.)

A.C. resistance of theconductor per unit lengthat 850C (micro ohm/cm.)

Conductor cross-sectionarea, cm2

Dielectric loss, W/cm.

Dimensions (Overall

Diameter), cm.

Conductor

Insulation

Sheath

Bedding

Serving (= Total)

Oil

Copper

Dielectric

Lead (sheath)

Cotton tape (bedding)Polythene (serving)

750

0.26

2.00.39

500 1.60

2.9 0.13

500

542 1.5

0.9

8.93

1.05

11.37

0.92

NOMENCLATURE

A Cross-sectional area of tunnel (m2).

s of F and p for different forma- Cp Specific heat (J/g °C).s Dt Diameter of tunnel (m).

0.785 (F), 0.21 (p); D diameter of cable (cm.).0.6 (F), 0.225 (p); F Ratio; Convective Heat Transferred

Total Heat Transferred

0.55, 0.2 (p); g Thermal resistivity (°C cm/W).0.40, 0.215 (p); H Thermal conductance per unit axial length (W/°C).0.48, 0.2 (p). H Thermal conductance between cable surface and air for axial

per cm length is shown to 0.618 ca length L of system (W/0C).Haw Thermal conductance between moving air and tunnel walls for

axial length L of system (W/°C).K Thermal conductivity (W/°C.cm.) - of air K; of soil KS, of con-

crete Kc.L Length (m).n Number of cables.

Heat per unit length disipated by radiation (W).

!Qc Heat per unit length disipated by convection (W).

Cables per length.

Cables (thermal resistances are per unit axial length).R Thermal resistance between cable surface to walls due to radia-

132 KV 275 KV 400 KV r tion (°C/W).Rca Thermal resistance between single cable surface and air by con-

vection (°C/W).0.19 0.099

0.96 0.25 0.124

2.90 9.65 19.35

0.0416 0.071 0.14

2.546 4.09 5.766

4.674 7.51 10.620

5.148 8.068 11.50

5.66 8.22 11.56

6.25 9.10 12.50

Rc' Thermal resistance between number of cable surfaces and air byconvection (°C/W).

Rc Thermal resistance between moving air and tunnel walls (°C/W).Rd Thermal resistance of cable dielectric (°C/W).Rb Thermal resistance of bedding material (°C/W).

Rser Thermal resistance of serving material (°C/W).aT Absolute temperature (K).T Temperature (deg. C).

Tsh Temperature of sheath.

Ti Temperature of ground surface.Tc Temperature of conductor.

Tser Temperature of serving.

TS Temperature of cable surface.Tw Temperature of tunnel walls.TaW Temperature of air at inlet to segment of tunnel.

Ta2 Temperature of air at outlet of segment of tunnel.

Twl Temperature of centre of tunnel walls.

306

Specific Densityheat gm/cm3J/°C. gmi.

TSa Bulk air temperature.Tsn Temperature of centre of nth annular cylinder of soil.t Time (s).At Time interval (s).

V Velocity (m/s).

Vm Air velocity at outlet (m/s).W Total heat generated from cables (W per unit length).0 Temperature rise (deg C).p Density (gm/cm3).ac Diffusivity (cm2/s).

REFERENCES

[1] Weedy, B.M. and El Zayyat H.M.: 'Heat transfer from cables intunnels and shafts'. Paper to I.E.E.E., Summer Meeting 1972.

[2] Giaro, J.A.: 'Temperature rise of power cables in a gallery withforced ventilation'. CIGRE, 1960, Paper No. 213.

[3] Kitagawa, K.: 'Forced cooling of power cables in Japan'. Its studiesand performance CIGRE, Paper No. 213, 1964.

[4] Burrell, R.W., Falcone, A.J. and Roberts, W.J.: 'Forced air coolingfor station cables'. Trans. A.I.E.E., III, 70, 1951, pp. 1798-1803.

[5] Germany, N.: 'Calculation of the temperature rise of cables in gal-lery with forced ventilation'. Rev. Electr., 1963, suppl. to Bull.Soc. Roy. Belge Electr. Brussels, 4, No. 1, pp. 3-13.

[6] Whitehead, S. and Hutchings, E.E.: 'Current ratings of cables fortransmission and distribution', Journal I.E.E., 1938, pp. 157-165.

[7] Van Wormer, F.C.: 'An improved approximate technique for calcu-lating temperature transients'. Trans. AIE.E., 74, 1955, pp. 277-280.

ACKNOWLEDGEMENTS

The authors wish to thank the University of Southampton for theprovision of computing and other facilities.

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